Killing–Hopf theorem

Characterizes complete connected Riemannian manifolds of constant curvature

In geometry, the Killing–Hopf theorem states that complete connected Riemannian manifolds of constant curvature are isometric to a quotient of a sphere, Euclidean space, or hyperbolic space by a group acting freely and properly discontinuously. These manifolds are called space forms. The Killing–Hopf theorem was proved by Killing (1891) and Hopf (1926).

References

Hopf, Heinz (1926), "Zum Clifford-Kleinschen Raumproblem", Mathematische Annalen, 95 (1): 313–339, doi:10.1007/BF01206614, ISSN 0025-5831
Killing, Wilhelm (1891), "Ueber die Clifford-Klein'schen Raumformen", Mathematische Annalen, 39 (2): 257–278, doi:10.1007/BF01206655, ISSN 0025-5831

Manifolds (Glossary)

Basic concepts

Topological manifold
- Atlas
Differentiable/Smooth manifold
- Differential structure
- Smooth atlas
Submanifold
Riemannian manifold
Smooth map
Submersion
Pushforward
Tangent space
Differential form
Vector field

Main results (list)

Maps

Curve
Diffeomorphism
- Local
Geodesic
Exponential map
- in Lie theory
Foliation
Immersion
Integral curve
Lie derivative
Section
Submersion

Types of
manifolds

Tensors

Vectors	Distribution Lie bracket Pushforward Tangent space bundle Torsion Vector field Vector flow
Covectors	Closed/Exact Covariant derivative Cotangent space bundle De Rham cohomology Differential form Vector-valued Exterior derivative Interior product Pullback Ricci curvature flow Riemann curvature tensor Tensor field density Volume form Wedge product
Bundles	Adjoint Affine Associated Cotangent Dual Fiber (Co) Fibration Jet Lie algebra (Stable) Normal Principal Spinor Subbundle Tangent Tensor Vector
Connections	Affine Cartan Ehresmann Form Generalized Koszul Levi-Civita Principal Vector Parallel transport